Subgroups — C₄×C₅²×AGL(1,𝔽₅) #14522783

Order 2000.

276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₄×C₅²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅²×D₅	C₂²	1	1	yes	characteristic
4	C₅²×AGL(1,𝔽₅)	C₄	1	4	yes	direct product factor
5	C₄×C₅³	C₄	1	1	yes	characteristic, complemented
6	C₅²×Dic₅	C₄	1	1	yes	characteristic, complemented
7	C₄×C₅×AGL(1,𝔽₅)	C₅	1	6	yes	maximal, direct product factor
8	C₄²×C₅²	—	5	1	no	maximal
9	C₂×C₅³	C₂×C₄	1	1	yes	characteristic
10	C₅²×D₅	C₂×C₄	1	2	yes	—
11	C₂×C₅×AGL(1,𝔽₅)	C₂×C₅	1	12	yes	—
12	C₂×C₄×C₅²	—	5	2	no	—
13	C₂×C₄×C₅²	—	5	1	no	—
14	C₄×C₅×D₅	C₂×C₅	1	6	yes	—
15	C₅³	C₄²	1	1	yes	characteristic, complemented, Sylow
16	C₂²×C₅²	—	5	1	no	—
17	C₂×C₅×D₅	C₂²×C₅	1	6	yes	—
18	C₅×AGL(1,𝔽₅)	C₄×C₅	1	24	yes	direct product factor
19	C₄×C₅²	—	4	6	no	—
20	C₄×C₅²	—	5	4	no	—
21	C₄×C₅²	—	5	1	no	—
22	C₄×C₅²	C₄×C₅	1	6	yes	complemented
23	C₄×C₅²	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
24	C₅×Dic₅	C₄×C₅	1	6	yes	complemented
25	C₄×AGL(1,𝔽₅)	C₅²	1	1	yes	characteristic, direct product factor
26	C₄²×C₅	—	5	6	no	—
27	C₂×C₅²	—	4	6	no	—
28	C₂×C₅²	—	5	2	no	—
29	C₂×C₅²	C₂×C₄×C₅	1	6	yes	—
30	C₂×C₅²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₅×D₅	C₂×C₄×C₅	1	12	yes	—
32	C₂×AGL(1,𝔽₅)	C₂×C₅²	1	2	yes	—
33	C₂×C₄×C₅	—	5	12	no	—
34	C₂×C₄×C₅	—	5	6	no	—
35	C₄×D₅	C₂×C₅²	1	1	yes	characteristic
36	C₅²	—	4	6	no	—
37	C₅²	C₄²×C₅	1	6	yes	complemented
38	C₅²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
39	C₂²×C₅	—	5	6	no	—
40	C₂×D₅	C₂²×C₅²	1	1	yes	characteristic
41	AGL(1,𝔽₅)	C₄×C₅²	1	4	yes	direct product factor
42	C₄×C₅	—	4	6	no	—
43	C₄×C₅	—	5	24	no	—
44	C₄×C₅	—	5	6	no	—
45	C₄×C₅	C₄×C₅²	1	1	yes	characteristic, complemented
46	C₄×C₅	C₅×AGL(1,𝔽₅)	1	6	yes	direct product factor
47	Dic₅	C₄×C₅²	1	1	yes	characteristic, complemented
48	C₄²	—	5	1	no	Sylow
49	C₂×C₅	—	4	6	no	—
50	C₂×C₅	—	5	12	no	—
51	C₂×C₅	C₂×C₄×C₅²	1	1	yes	characteristic
52	C₂×C₅	C₂×C₅×AGL(1,𝔽₅)	1	6	yes	—
53	D₅	C₂×C₄×C₅²	1	2	yes	—
54	C₂×C₄	—	5	2	no	—
55	C₂×C₄	—	5	1	no	—
56	C₅	—	4	6	no	—
57	C₅	C₄²×C₅²	1	1	yes	characteristic, complemented
58	C₅	C₄×C₅×AGL(1,𝔽₅)	1	6	yes	direct product factor
59	C₂²	—	5	1	no	—
60	C₄	—	5	4	no	—
61	C₄	—	5	1	no	—
62	C₄	C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor
63	C₂	—	5	2	no	—
64	C₂	C₂×C₅²×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₅²	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₄²×C₅²
Frattini	C₂	C₂×C₅²×AGL(1,𝔽₅)
Fitting	C₄×C₅³	C₄
socle	C₂×C₅³	C₂×C₄
prime radical	C₂×C₅²×D₅	C₂²
cosocle	C₂×D₅	C₂²×C₅²
LCS	G ▷ C₅
UCS	C₁ ◁ C₄×C₅²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₅²×AGL(1,𝔽₅)	C₄	direct (N × H)
5. C₄×C₅³	C₄	semidirect
6. C₅²×Dic₅	C₄	semidirect
7. C₄×C₅×AGL(1,𝔽₅)	C₅	direct (N × H)
15. C₅³	C₄²	semidirect
18. C₅×AGL(1,𝔽₅)	C₄×C₅	direct (N × H)
22. C₄×C₅²	C₄×C₅	semidirect
23. C₄×C₅²	AGL(1,𝔽₅)	direct (N × H)
24. C₅×Dic₅	C₄×C₅	semidirect
25. C₄×AGL(1,𝔽₅)	C₅²	direct (N × H)
37. C₅²	C₄²×C₅	semidirect
38. C₅²	C₄×AGL(1,𝔽₅)	direct (N × H)
41. AGL(1,𝔽₅)	C₄×C₅²	direct (N × H)
45. C₄×C₅	C₄×C₅²	semidirect
46. C₄×C₅	C₅×AGL(1,𝔽₅)	direct (N × H)
47. Dic₅	C₄×C₅²	semidirect
57. C₅	C₄²×C₅²	semidirect
58. C₅	C₄×C₅×AGL(1,𝔽₅)	direct (N × H)
62. C₄	C₅²×AGL(1,𝔽₅)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅²×AGL(1,𝔽₅)	2
C₄×C₅²×D₅	2
C₄×C₅×AGL(1,𝔽₅)	5
C₄²×C₅²	5

Subgroups — C4×C52×AGL(1,𝔽5) #14522783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅²×AGL(1,𝔽₅) #14522783